In quantum mechanics, the position and momentum of a particle cannot be known simultaneously. In special relativity, the idea of simultaneity loses its meaning. Would it be possible to measure first the position of a particle and then the momentum of a particle, and do a Lorentz transformation into a frame where those two measurements are made simultaneously? If not, why not? If that is possible, does this violate the Heisenberg uncertainty principle?
Furthermore, would it be possible for an observer who makes those two measurements at different points in time to transmit that information into a Lorentz frame in which those two measurements are made simultaneously?
Best Answer
Firstly, it is not that the position and the momentum of a system cannot be measured by an apparatus simultaneously in time. It can be, only that their standard deviations will have to satisfy the uncertainity principle. Here the fact that in QM the position and momentum of a particle cannot be known simultaneously means that position and momentum operators acting on a system don't have simultaneous eigenstates, since they don't commute. This has no reference with time-simultaneity in relativity. Please see this answer too for more details Simultaneously measurement in quantum mechanics?,
In practice for any measurement we make, we always make a sort of a "combined" measurement. We never use infinitely large detectors for measuring momentum and similar cases for measuring position. So simultaneous measurements in time can always be made, but simultaneous measurements of eigenkets of non-commuting observables cannot be made.